What happens if you just keep squaring?

A simple “keep squaring” pattern leads to a number that is equal to its own square—an object with infinitely many digits to the left of the decimal point. Starting from 5, then 25, then 625, the last digits keep matching the previous number for longer stretches: squaring 390,625 preserves the final 5 digits, squaring the matching tail preserves more, and the shared suffix grows without end. That iterative stabilization behaves like a convergence, but in a different direction than ordinary real-number limits. The result is a 10-adic number n satisfying n² = n, meaning n is its own square.

The central payoff is that these “left-infinite” numbers are not mathematical nonsense; they live in a different number system—p-adic arithmetic—where distance and size are defined by how far two numbers agree in their base-p expansions. In 10-adics, addition and multiplication work digit-by-digit from the right, because each new digit depends only on digits already fixed to the right. This makes it possible to represent fractions without writing division symbols. For instance, multiplying a specific 10-adic number ending in 857142857143 by 7 yields 1, so that number equals 1/7. Similarly, 1/3 can be encoded as an infinite string of digits that, when multiplied by 3, produces a 1 in the units place and zeros everywhere else—an analogue of the familiar 0.999… = 1 argument, but flipped to the left.

The transcript then tackles why p-adics are powerful despite looking alien. In ordinary arithmetic, factoring relies on the idea that if a product is zero, at least one factor must be zero. That property fails in base 10 because 10 is composite (5×2), allowing nonzero 10-adic numbers to multiply to zero via digit-carry interactions. The fix is to switch to a prime base p. In p-adics, the “zero product” behavior returns: a product of nonzero p-adic numbers cannot be zero. This prime-base structure is what makes p-adics a reliable tool for solving equations.

A major example comes from Diophantine problems tied to Fermat’s Last Theorem. The transcript uses Kurt Hensel’s method: build solutions as expansions in increasing powers of a prime (like 3), determining coefficients one modulus at a time (mod 3, then mod 9, then mod 27, and so on). Applied to a “sum of squares” equation derived from Diophantus, the digit-by-digit lifting process produces a 3-adic solution whose base-3 expansion is an infinite string of 1s. Even though that expansion makes no sense as a real number, p-adic geometry interprets it correctly: the infinite series behaves like a geometric series with ratio 3, yet it still yields a finite p-adic value. The result corresponds to a rational solution—specifically, a rational x = -1/2 that generates a square-area identity.

Finally, the transcript connects the abstract machinery to real breakthroughs: Wiles’s proof of Fermat’s Last Theorem used p-adic methods and famously required a “three, five trick,” switching from prime 3 to prime 5 when the prime-3 approach hit obstacles. The takeaway is that p-adics reshape notions of size and closeness so drastically that problems impossible over the reals become tractable—turning “infinite left digits” into a practical computational framework for modern number theory.